Average Error: 38.6 → 23.2
Time: 4.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.68133238957017115 \cdot 10^{161}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{1}}{\frac{-1 \cdot re - re}{im}}}\\ \mathbf{elif}\;re \le 1.4937844730964523 \cdot 10^{-255}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|im\right| \cdot \frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} - re}\right)}\\ \mathbf{elif}\;re \le 2.37750825737911996 \cdot 10^{102}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.68133238957017115 \cdot 10^{161}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{1}}{\frac{-1 \cdot re - re}{im}}}\\

\mathbf{elif}\;re \le 1.4937844730964523 \cdot 10^{-255}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|im\right| \cdot \frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} - re}\right)}\\

\mathbf{elif}\;re \le 2.37750825737911996 \cdot 10^{102}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} + re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\

\end{array}
double f(double re, double im) {
        double r192430 = 0.5;
        double r192431 = 2.0;
        double r192432 = re;
        double r192433 = r192432 * r192432;
        double r192434 = im;
        double r192435 = r192434 * r192434;
        double r192436 = r192433 + r192435;
        double r192437 = sqrt(r192436);
        double r192438 = r192437 + r192432;
        double r192439 = r192431 * r192438;
        double r192440 = sqrt(r192439);
        double r192441 = r192430 * r192440;
        return r192441;
}
↓
double f(double re, double im) {
        double r192442 = re;
        double r192443 = -1.6813323895701712e+161;
        bool r192444 = r192442 <= r192443;
        double r192445 = 0.5;
        double r192446 = 2.0;
        double r192447 = im;
        double r192448 = 1.0;
        double r192449 = pow(r192447, r192448);
        double r192450 = -1.0;
        double r192451 = r192450 * r192442;
        double r192452 = r192451 - r192442;
        double r192453 = r192452 / r192447;
        double r192454 = r192449 / r192453;
        double r192455 = r192446 * r192454;
        double r192456 = sqrt(r192455);
        double r192457 = r192445 * r192456;
        double r192458 = 1.4937844730964523e-255;
        bool r192459 = r192442 <= r192458;
        double r192460 = fabs(r192447);
        double r192461 = r192442 * r192442;
        double r192462 = r192447 * r192447;
        double r192463 = r192461 + r192462;
        double r192464 = sqrt(r192463);
        double r192465 = r192464 - r192442;
        double r192466 = r192460 / r192465;
        double r192467 = r192460 * r192466;
        double r192468 = r192446 * r192467;
        double r192469 = sqrt(r192468);
        double r192470 = r192445 * r192469;
        double r192471 = 2.37750825737912e+102;
        bool r192472 = r192442 <= r192471;
        double r192473 = log(r192464);
        double r192474 = exp(r192473);
        double r192475 = r192474 + r192442;
        double r192476 = r192446 * r192475;
        double r192477 = sqrt(r192476);
        double r192478 = r192445 * r192477;
        double r192479 = 2.0;
        double r192480 = r192479 * r192442;
        double r192481 = r192446 * r192480;
        double r192482 = sqrt(r192481);
        double r192483 = r192445 * r192482;
        double r192484 = r192472 ? r192478 : r192483;
        double r192485 = r192459 ? r192470 : r192484;
        double r192486 = r192444 ? r192457 : r192485;
        return r192486;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.6
Target33.8
Herbie23.2
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if re < -1.6813323895701712e+161

    1. Initial program 64.0

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
    2. Using strategy rm

    3. Applied flip-+64.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    4. Simplified51.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
    5. Using strategy rm

    6. Applied sqr-pow51.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{\left(\frac{2}{2}\right)} \cdot {im}^{\left(\frac{2}{2}\right)}}}{\sqrt{re \cdot re + im \cdot im} - re}}\]

    7. Applied associate-/l*51.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{{im}^{\left(\frac{2}{2}\right)}}{\frac{\sqrt{re \cdot re + im \cdot im} - re}{{im}^{\left(\frac{2}{2}\right)}}}}}\]

    8. Simplified51.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{\left(\frac{2}{2}\right)}}{\color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} - re}{im}}}}\]

    9. Taylor expanded around -inf 23.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{\left(\frac{2}{2}\right)}}{\frac{\color{blue}{-1 \cdot re} - re}{im}}}\]

    if -1.6813323895701712e+161 < re < 1.4937844730964523e-255

    1. Initial program 38.9

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
    2. Using strategy rm

    3. Applied flip-+38.9

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    4. Simplified31.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity31.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\color{blue}{1 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}}\]

    7. Applied add-sqr-sqrt31.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{\sqrt{{im}^{2}} \cdot \sqrt{{im}^{2}}}}{1 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}\]

    8. Applied times-frac31.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{\sqrt{{im}^{2}}}{1} \cdot \frac{\sqrt{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} - re}\right)}}\]

    9. Simplified30.8

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left|im\right|} \cdot \frac{\sqrt{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} - re}\right)}\]

    10. Simplified29.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left|im\right| \cdot \color{blue}{\frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} - re}}\right)}\]

    if 1.4937844730964523e-255 < re < 2.37750825737912e+102

    1. Initial program 19.3

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
    2. Using strategy rm

    3. Applied add-exp-log21.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}} + re\right)}\]

    if 2.37750825737912e+102 < re

    1. Initial program 52.7

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

    2. Taylor expanded around inf 10.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(2 \cdot re\right)}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification23.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.68133238957017115 \cdot 10^{161}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{1}}{\frac{-1 \cdot re - re}{im}}}\\ \mathbf{elif}\;re \le 1.4937844730964523 \cdot 10^{-255}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|im\right| \cdot \frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} - re}\right)}\\ \mathbf{elif}\;re \le 2.37750825737911996 \cdot 10^{102}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))